Vector rotation around X-axis

Algebra Level pending

If the vector V ⃗ = [ 2 , 5 , 3 ] T \vec {V} = [2, 5, 3 ]^T rotates − 9 0 ∘ -90^\circ about the x x -axis, then the rotated vector can be expressed as __________ . \text{\_\_\_\_\_\_\_\_\_\_}.

[ 2 , 5 , − 3 ] T [2, 5, -3 ]^T [ 2 , 3 , − 5 ] T [2, 3, -5 ]^T [ 2 , − 3 , − 5 ] T [2, -3, -5 ]^T [ 3 , − 2 , − 5 ] T [3, -2, -5 ]^T

Ossama Ismail by Ossama Ismail , 63, Egypt

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1 solution

Ossama Ismail
Mar 23, 2017

Also you can solve this problem using the 3 × 3 3 \times 3 rotation matrix about X-axis by angle θ \theta :

R x , θ = ( 1 0 0 0 cos ⁡ ( θ ) − sin ⁡ ( θ ) 0 sin ⁡ ( θ ) cos ⁡ ( θ ) ) = ( 1 0 0 0 0 1 0 − 1 0 ) Rotated vector = [ R x , − 90 ] T . V ⃗ = ( 1 0 0 0 0 − 1 0 1 0 ) . ( 2 5 3 ) = ( 2 3 − 5 ) \begin{aligned} R_{x,\theta} &= \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & \cos(\theta) &-\sin(\theta) \\ 0 & \sin(\theta) &\cos(\theta) \\ \end{array}\right) = \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 &0 &1 \\ 0 & -1 &0\\ \end{array}\right) \\ \\ \\ \text{Rotated vector } &= {[R_{x,-90}]}^T . \vec V = \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 &0 &-1 \\ 0 & 1 &0\\ \end{array}\right) . \left(\begin{array}{c} 2\\ 5 \\ 3\\ \end{array}\right) = \left(\begin{array}{c} 2\\ 3 \\ -5\\ \end{array}\right) \end{aligned}

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